Avoiding chaos

Complex interactions can create persistent fluctuations in high-diversity ecosystems

When can ecological interactions drive an entire ecosystem into a persistent non-equilibrium state, where many species populations fluctuate without going to extinction? We show that high-diversity spatially heterogeneous systems can exhibit chaotic dynamics which persist for extremely long times. We develop a theoretical framework, based on dynamical mean-field theory, to quantify the conditions under which these fluctuating states exist, and predict their properties. We uncover parallels with the persistence of externally-perturbed ecosystems, such as the role of perturbation strength, synchrony and correlation time. But uniquely to endogenous fluctuations, these properties arise from the species dynamics themselves, creating feedback loops between perturbation and response. A key result is that fluctuation amplitude and species diversity are tightly linked: in particular, fluctuations enable dramatically more species to coexist than at equilibrium in the very same system. Our findings highlight crucial differences between well-mixed and spatially-extended systems, with implications for experiments and their ability to reproduce natural dynamics. They shed light on the maintenance of biodiversity, and the strength and synchrony of fluctuations observed in natural systems.

Dynamical properties of maps fitted to data in the noise-free limit

We argue that any attempt to classify dynamical properties from nonlinear finite time-series data requires a mechanistic model fitting the data better than piecewise linear models according to standard model selection criteria. Such a procedure seems necessary but still not sufficient.

Chaos reduces species extinction by amplifying local population noise

In the mid-1970s, theoretical ecologists were responsible for stimulating interest in nonlinear dynamics and chaos. Ironically, the importance of chaos in ecology itself remains controversial. Proponents of ecological chaos point to its ubiquity in mathematical models and to various empirical findings. Sceptics maintain that the models are unrealistic and that the experimental evidence is equally consistent with stochastic models. More generally, it has been argued that interdemic selection and/or enhanced rates of species extinction will eliminate populations and species that evolve into chaotic regions of parameter space. Fundamental to this opinion is the belief that violent oscillations and low minimum population densities are inevitable correlates of the chaotic state. In fact, rarity is not a necessary consequence of complex dynamical behaviour. But even when chaos is associated with frequent rarity, its consequences to survival are necessarily deleterious only in the case of species composed of a single population. Of course, the majority of real world species (for example, most insects) consist of multiple populations weakly coupled by migration, and in this circumstance chaos can actually reduce the probability of extinction. Here we show that although low densities lead to more frequent extinction at the local level, the decorrelating effect of chaotic oscillations reduces the degree of synchrony among populations and thus the likelihood that all are simultaneously extinguished.